A linear transformation is given by multiplying by its matrix representation with respect to bases of the spaces

Created almost 9 years ago, updated about 8 hours ago

If $A=[T]_{C\leftarrow B}$, then $[T(v)]_C=A[v]_B$. Expand $v$ in basis $B$, apply linearity, and collect coefficients in basis $C$. The product is an output coordinate vector, not necessarily the abstract output vector itself.

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